Properties

Label 2-3200-40.19-c0-0-5
Degree $2$
Conductor $3200$
Sign $-0.447 - 0.894i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 1.99·9-s − 1.73·11-s + i·17-s − 1.73·19-s + 1.73i·27-s + 2.99i·33-s − 41-s − 49-s + 1.73·51-s + 2.99i·57-s − 1.73i·67-s i·73-s + 0.999·81-s + 1.73i·83-s + ⋯
L(s)  = 1  − 1.73i·3-s − 1.99·9-s − 1.73·11-s + i·17-s − 1.73·19-s + 1.73i·27-s + 2.99i·33-s − 41-s − 49-s + 1.73·51-s + 2.99i·57-s − 1.73i·67-s i·73-s + 0.999·81-s + 1.73i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1909982403\)
\(L(\frac12)\) \(\approx\) \(0.1909982403\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.263351584569671659079884194421, −7.66559062196886698524791956676, −6.84887569111194782469599435438, −6.23404515549729927076674638751, −5.55821380405303154270464010164, −4.59004349436226075542588884414, −3.25668123468542451196049438812, −2.32326956396262588260463321118, −1.70215958604729267975548296727, −0.10337546939216796599228451727, 2.35749732806830664068271494801, 3.05909424569579090849823202821, 4.02249497625762682181511290640, 4.80596252749770486759981497688, 5.24116864240490708735342503743, 6.09279481991954981275129654767, 7.16124309371047992442225908326, 8.187190993340122413050511981255, 8.630020151629887469556433109339, 9.490500766666234831839415075838

Graph of the $Z$-function along the critical line